Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process

TL;DR

This paper establishes a connection between noncolliding Brownian motion with drift and biorthogonal Stieltjes-Wigert matrix models, providing explicit probability densities, determinantal structure, and correlation kernels, with implications for Chern-Simons theory.

Contribution

It introduces a multitime joint probability density for noncolliding Brownian motion with drift and links it to biorthogonal Stieltjes-Wigert matrix models, revealing their determinantal structure.

Findings

The probability density transforms into a biorthogonal Stieltjes-Wigert matrix model.

The process is shown to be a determinantal point process at each time.

Explicit correlation kernels are derived for the process.

Abstract

Using the determinantal formula of Biane, Bougerol, and O'Connell, we give multitime joint probability densities to the noncolliding Brownian motion with drift, where the number of particles is finite. We study a special case such that the initial positions of particles are equidistant with a period $a$ and the values of drift coefficients are well-ordered with a scale $\sigma$. We show that, at each time $t >0$, the single-time probability density of particle system is exactly transformed to the biorthogonal Stieltjes-Wigert matrix model in the Chern-Simons theory introduced by Dolivet and Tierz. Here one-parameter extensions ($\theta$-extensions) of the Stieltjes-Wigert polynomials, which are themselves $q$-extensions of the Hermite polynomials, play an essential role. The two parameters $a$ and $\sigma$ of the process combined with time $t$ are mapped to the parameters $q$ and $\theta$ of the biorthogonal polynomials. By the transformation of normalization factor of our probability density, the partition function of the Chern-Simons matrix model is readily obtained. We study the determinantal structure of the matrix model and prove that, at each time $t >0$, the present noncolliding Brownian motion with drift is a determinantal point process, in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel. Using the obtained correlation kernel, we study time evolution of the noncolliding Brownian motion with drift.